Sparse covariance matrices arise naturally in a range of applications including genomics. Estimation of sparse covariance matrices has been considered in several recent papers in the setting of complete data (see, e.g., Bickel and Levina [1]}, El Karoui [2]}, Rothman et al. [3]}, Cai and Zhou [4]} and Cai and Liu [5]}). Estimation of a sparse covariance matrix is intrinsically a heteroscedastic problem in the sense that the variances of the entries of the sample covariance matrix can vary over a wide range. To treat the heteroscedasticity of the sample covariances, Cai and Liu [5]} introduced an adaptive thresholding procedure which adapts to the variability of the individual entries of the sample covariance matrix and outperforms the universal thresholding method. The estimator is shown to be simultaneously rate optimal over collections of sparse covariance matrices.
